When test forms are calibrated separately, item response theory parameters are not comparable because they are expressed on different measurement scales. The equating process includes the conversion of item parameter estimates on a common scale and the determination of comparable test scores. Various statistical methods have been proposed to perform equating between two test forms. This paper provides a generalization to multiple test forms of the mean-geometric mean, the mean-mean, the Haebara, and the Stocking–Lord methods. The proposed methods estimate simultaneously the equating coefficients that permit the scale transformation of the parameters of all forms to the scale of the base form. Asymptotic standard errors of the equating coefficients are derived. A simulation study is presented to illustrate the performance of the methods.

The psychometric process used to establish a relationship between the scores of two (or more) instruments is generically referred to as linking. When two instruments with the same content and statistical test specifications are linked, these instruments are said to be equated. Linking and equating procedures have long been used for practical benefit in educational testing. In recent years, health outcome researchers have increasingly applied linking techniques to patient-reported outcome (PRO) data. However, these applications have some noteworthy purposes and associated methodological questions. Purposes for linking health outcomes include the harmonization of data across studies or settings (enabling increased power in hypothesis testing), the aggregation of summed score data by means of score crosswalk tables, and score conversion in clinical settings where new instruments are introduced, but an interpretable connection to historical data is needed. When two PRO instruments are linked, assumptions for equating are typically not met and the extent to which those assumptions are violated becomes a decision point around how (and whether) to proceed with linking. We demonstrate multiple linking procedures—equipercentile, unidimensional IRT calibration, and calibrated projection—with the Patient-Reported Outcomes Measurement Information System Depression bank and the Patient Health Questionnaire-9. We validate this link across two samples and simulate different instrument correlation levels to provide guidance around which linking method is preferred. Finally, we discuss some remaining issues and directions for psychometric research in linking PRO instruments.

The PROsetta Stone Project, summarized in this issue by Schalet et al. (Psychometrika 86, 2021), is a major step forward in enabling comparability between different patient-reported outcomes measures. Schalet et al. clearly describe the psychometric methods used in the PROsetta Stone project and other projects from the Patient-Reported Outcomes Measurement Information System (PROMIS): linking based on unidimensional item response theory (IRT), equipercentile linking, and calibrated projection based on multidimensional IRT. Analyses in a validation data set and simulation studies provide strong support that the linking methods are robust when basic assumptions are fulfilled. The links already established will be of great value to the field, and the methodology described by Schalet et al. will hopefully inspire the next series of linking studies. Among potential improvements that should be considered by new studies are: (1) a thorough evaluation of the content of the measures to be linked to better guide the evaluation of measurement assumptions, (2) improvements in the design of linking studies such as selection of the optimal sample to provide data in the score ranges where linking precision is most critical and using counterbalanced designs to control for order effects. Finally, it may be useful to consider how the linking algorithms are used in subsequent data analyses. Analytic strategies based on plausible values or latent regression IRT models may be preferable to the simple transformation of scores from one patient at the time.

The $\theta$\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\theta $$\end{document} metric in item response theory is often not the most useful metric for score reporting or interpretation. In this paper, I demonstrate that the filtered monotonic polynomial (FMP) item response model, a recently proposed nonparametric item response model (Liang & Browne in J Educ Behav Stat 40:5–34, 2015), can be used to specify item response models on metrics other than the $\theta$\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\theta $$\end{document} metric. Specifically, I demonstrate that any item response function (IRF) defined within the FMP framework can be re-expressed as another FMP IRF by taking monotonic transformations of the latent trait. I derive the item parameter transformations that correspond to both linear and nonlinear transformations of the latent trait metric. These item parameter transformations can be used to define an item response model based on any monotonic transformation of the $\theta$\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\theta $$\end{document} metric, so long as the metric transformation is approximated by a monotonic polynomial. I demonstrate this result by defining an item response model directly on the approximate true score metric and discuss the implications of metric transformations for applied testing situations.

Lord and Wingersky’s (Appl Psychol Meas 8:453–461, 1984) recursive algorithm for creating summed score based likelihoods and posteriors has a proven track record in unidimensional item response theory (IRT) applications. Extending the recursive algorithm to handle multidimensionality is relatively simple, especially with fixed quadrature because the recursions can be defined on a grid formed by direct products of quadrature points. However, the increase in computational burden remains exponential in the number of dimensions, making the implementation of the recursive algorithm cumbersome for truly high-dimensional models. In this paper, a dimension reduction method that is specific to the Lord–Wingersky recursions is developed. This method can take advantage of the restrictions implied by hierarchical item factor models, e.g., the bifactor model, the testlet model, or the two-tier model, such that a version of the Lord–Wingersky recursive algorithm can operate on a dramatically reduced set of quadrature points. For instance, in a bifactor model, the dimension of integration is always equal to 2, regardless of the number of factors. The new algorithm not only provides an effective mechanism to produce summed score to IRT scaled score translation tables properly adjusted for residual dependence, but leads to new applications in test scoring, linking, and model fit checking as well. Simulated and empirical examples are used to illustrate the new applications.

We establish some existence results for a class of critical $N$-Laplacian problems in a bounded domain in $\mathbb {R}^{N}$. In the absence of a suitable direct sum decomposition of the underlying Sobolev space to which the classical linking theorem can be applied, we use an abstract linking theorem based on the $\mathbb {Z}_2$-cohomological index to obtain a non-trivial critical point.

This article aims to test whether the Theta System of Reinhart (1996, 2000, 2001, 2002) can account for the puzzles associated with psychological verbs in Polish. The first puzzle, called argument linking, relates to the mapping of the Experiencer onto a subject or an object position. The second puzzle, referred to as case linking, concerns the fact that Experiencers may be marked for different cases in the same sentence position. The analysis of Object Experiencer (OE)/Subject Experiencer (SE) alternations in Polish carried out in this article demonstrates that the predictions of the Theta System about Experiencer argument linking are borne out by the Polish data. SE alternants of eventive OE verbs in Polish show unergative properties, which directly follows from the mechanisms of the Theta System. However, the Theta System faces problems when confronted with dyadic OE verbs with dative Experiencers. The model predicts that dative Experiencers are merged internally, as a part of an unaccusative structure. This prediction is untenable for Polish, because dative Experiencers of dyadic predicates show some characteristics of external arguments, and hence must merge externally. Consequently, the conclusion drawn is that the Theta System can provide solutions to some, but not all, of the argument and case-linking puzzles associated with Polish Experiencers.

The set N of all null geodesics of a globally hyperbolic $(d + 1)$-dimensional spacetime $(M, g)$ is naturally a smooth $(2d - 1)$-dimensional contact manifold. The sky of an event x in M is the subset X of N consisting of all null geodesics through x, and is an embedded Legendrian submanifold of N diffeomorphic to $S^{(d - 1)}$. It was conjectured by Low that for $d = 2$ two events x and y are causally related if and only if X and Y are linked (in an appropriate sense). We use the contact structure and knot polynomial calculations to prove this conjecture in certain particular cases, and suggest that for $d = 3$ smooth linking should be replaced with Legendrian linking.

In this paper we establish a variant and generalized weak linking theorem, which contains more delicate result and insures the existence of bounded Palais–Smale sequences of a strongly indefinite functional. The abstract result will be used to study the semilinear Schrödinger equation $-\Delta u+V(x)u=K(x)|u|^{2^\ast-2}u+g(x, u), u\in W^{1,2}({\bf R}^N)$, where N ≥ 4; V,K,g are periodic in xj for 1 ≤ j ≤ N and 0 is in a gap of the spectrum of -Δ + V; K>0. If $0<g(x, u)u\leq c|u|^{2^\ast}$ for an appropriate constant c, we show that this equation has a nontrivial solution.

In this paper we complete two tasks. First we extend the nonsmooth critical point theory of Chang to the case where the energy functional satisfies only the weaker nonsmooth Cerami condition and we also relax the boundary conditions. Then we study semilinear and quasilinear equations (involving the p-Laplacian). Using a variational approach we establish the existence of one and of multiple solutions. In simple existence theorems, we allow the right hand side to be discontinuous. In that case in order to have an existence theory, we pass to a multivalued approximation of the original problem by, roughly speaking, filling in the gaps at the discontinuity points.